This presentation is about financial statement

financial

1. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
1
Chapter 6
Multivariate models

2. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Simultaneous Equations Models
•All the models we have looked at thus far have been single equations models of the
form y = X + u
•All of the variables contained in the X matrix are assumed to be EXOGENOUS.
•y is an ENDOGENOUS variable.
 
An example from economics to illustrate - the demand and supply of a good:
 (1)
(2)
(3)
 where = quantity of the good demanded
= quantity of the good supplied
S
t
= price of a substitute good
T
t
= some variable embodying the state of technology
Q PSu
dt t t t

Q PTv
st t t t

QQ
dt st

Q
dt
Q
st

3. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Assuming that the market always clears, and dropping the time subscripts for
simplicity
(4)
(5)
This is a simultaneous STRUCTURAL FORM of the model.
 
•The point is that price and quantity are determined simultaneously (price
affects quantity and quantity affects price).
 
•P and Q are endogenous variables, while S and T are exogenous.
 
•We can obtain REDUCED FORM equations corresponding to (4) and (5) by
solving equations (4) and (5) for P and for Q (separately).
Simultaneous Equations Models:
The Structural Form
Q PSu
Q PTv

4. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Solving for Q,
(6)
 
•Solving for P,
(7)
 
•Rearranging (6),
 
 
 
(8)
 
Obtaining the Reduced Form
PSuPTv
Q SuQ Tv










PP TSvu
( )( ) () P TSvu
P T S
vu

















5. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Multiplying (7) through by ,
 
 
 
 
(9)
 
 
•(8) and (9) are the reduced form equations for P and Q.
Obtaining the Reduced Form (cont’d)
Q SuQ Tv  
QQ T Suv  
( )( ) ( )    Q T Suv
Q T S
uv



















6. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
 
•But what would happen if we had estimated equations (4) and (5), i.e. the
structural form equations, separately using OLS?
 
•Both equations depend on P. One of the CLRM assumptions was that
E(Xu) = 0, where X is a matrix containing all the variables on the RHS of
the equation.
 
•It is clear from (8) that P is related to the errors in (4) and (5) - i.e. it is
stochastic.
 
•What would be the consequences for the OLS estimator, , if we ignore
the simultaneity?
 
Simultaneous Equations Bias



7. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Recall that and
 
•So that
 
•Taking expectations,
 
•If the X’s are non-stochastic, E(Xu) = 0, which would be the case in a single
equation system, so that , which is the condition for unbiasedness.
•But .... if the equation is part of a system, then E(Xu)  0, in general.
Simultaneous Equations Bias (cont’d)

(')'

XXXy
1
yXu
E E EXXXu
XXEXu
(

)()((')')
(')(')
 

 



1
1
E(

)
uXXX
uXXXXXXX
uXXXX
')'(
')'(')'(
)(')'(ˆ
1
11
1













8. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Conclusion: Application of OLS to structural equations which are part
of a simultaneous system will lead to biased coefficient estimates.
 
•Is the OLS estimator still consistent, even though it is biased?
•No - In fact the estimator is inconsistent as well.
•Hence it would not be possible to estimate equations (4) and (5)
validly using OLS.
Simultaneous Equations Bias (cont’d)

9. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
So What Can We Do?
•Taking equations (8) and (9), we can rewrite them as
(10)
 
(11)
 
•We CAN estimate equations (10) & (11) using OLS since all the RHS
variables are exogenous.
 
•But ... we probably don’t care what the values of the  coefficients
are; what we wanted were the original parameters in the structural
equations - , , , , , .
Avoiding Simultaneous Equations Bias
P T S  
10 11 12 1
Q T S  
20 21 22 2

10. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Can We Retrieve the Original Coefficients from the ’s?
Short answer: sometimes.
 
•As well as simultaneity, we sometimes encounter another problem:
identification.
•Consider the following demand and supply equations
 Supply equation(12)
Demand equation(13)
 We cannot tell which is which!
•Both equations are UNIDENTIFIED or NOT IDENTIFIED, or
UNDERIDENTIFIED.
•The problem is that we do not have enough information from the equations to
estimate 4 parameters. Notice that we would not have had this problem with
equations (4) and (5) since they have different exogenous variables.
Identification of Simultaneous Equations
Q P
Q P

11. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•We could have three possible situations:
 
1. An equation is unidentified
·
    like (12) or (13)
·
    we cannot get the structural coefficients from the reduced form estimates
 
2. An equation is exactly identified
·
    e.g. (4) or (5)
·
    can get unique structural form coefficient estimates
 
3. An equation is over-identified
·
    Example given later
·
    More than one set of structural coefficients could be obtained from the reduced form.
 
What Determines whether an Equation is Identified
or not?

12. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•How do we tell if an equation is identified or not?
•There are two conditions we could look at:
 
- The order condition - is a necessary but not sufficient condition for an
equation to be identified.
 
- The rank condition - is a necessary and sufficient condition for
identification. We specify the structural equations in a matrix form and
consider the rank of a coefficient matrix.
What Determines whether an Equation is Identified
or not? (cont’d)

13. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Statement of the Order Condition (from Ramanathan 1995, pp.666)
•Let G denote the number of structural equations. An equation is just
identified if the number of variables excluded from an equation is G-1.
 
•If more than G-1 are absent, it is over-identified. If less than G-1 are
absent, it is not identified.
 
Example
•In the following system of equations, the Y’s are endogenous, while the
X’s are exogenous. Determine whether each equation is over-, under-, or
just-identified.
 
(14)-(16)
Simultaneous Equations Bias (cont’d)
Y Y Y X Xu
Y Y Xu
Y Yu
1 0 12 33 41 52 1
2 0 13 21 2
3 0 12 3
    
  
 




14. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
 Solution
 
G = 3;
If # excluded variables = 2, the eq
n
is just identified
If # excluded variables > 2, the eq
n
is over-identified
If # excluded variables < 2, the eq
n
is not identified
 
Equation 14: Not identified
Equation 15: Just identified
Equation 16: Over-identified
Simultaneous Equations Bias (cont’d)

15. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•How do we tell whether variables really need to be treated as endogenous or not?
•Consider again equations (14)-(16). Equation (14) contains Y
2
and Y
3
- but do we
really need equations for them?
•We can formally test this using a Hausman test, which is calculated as follows:
1. Obtain the reduced form equations corresponding to (14)-(16). The reduced
forms turn out to be:
 
(17)-(19)
 
Estimate the reduced form equations (17)-(19) using OLS, and obtain the fitted
values,

Tests for Exogeneity
Y X Xv
Y X v
Y X v
1 10 111 122 1
2 20 211 2
3 30 311 3
  
 
 
 


,,YYY
123

16. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
 
2. Run the regression corresponding to equation (14).
 
3. Run the regression (14) again, but now also including the fitted values
as additional regressors:
 

(20)
 
4. Use an F-test to test the joint restriction that 
2
= 0, and 
3
= 0. If the
null hypothesis is rejected, Y
2
and Y
3
should be treated as endogenous.
Tests for Exogeneity (cont’d)
,YY
23
1
1
33
1
222514332101
ˆˆ
uYYXXYYY  

17. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Consider the following system of equations:
 
(21-23)
 
•Assume that the error terms are not correlated with each other. Can we estimate the equations
individually using OLS?
 
•Equation 21: Contains no endogenous variables, so X
1
and X
2
are not correlated with u
1
. So
we can use OLS on (21).
•Equation 22: Contains endogenous Y
1
together with exogenous X
1
and X
2
. We can use OLS
on (22) if all the RHS variables in (22) are uncorrelated with that equation’s error term. In
fact, Y
1
is not correlated with u
2
because there is no Y
2
term in equation (21). So we can use
OLS on (22).
Recursive Systems
Y X Xu
Y Y X Xu
Y Y Y X Xu
1 10 111 122 1
2 20 211 211222 2
3 30 311 322 311 322 3
   
   
    
  
 
 

18. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Equation 23: Contains both Y
1
and Y
2
; we require these to be
uncorrelated with u
3
. By similar arguments to the above, equations
(21) and (22) do not contain Y
3
, so we can use OLS on (23).
 
•This is known as a RECURSIVE or TRIANGULAR system. We do
not have a simultaneity problem here.
 
•But in practice not many systems of equations will be recursive...
Recursive Systems (cont’d)

19. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Cannot use OLS on structural equations, but we can validly apply it to
the reduced form equations.
 
•If the system is just identified, ILS involves estimating the reduced
form equations using OLS, and then using them to substitute back to
obtain the structural parameters.
 
•However, ILS is not used much because
1. Solving back to get the structural parameters can be tedious.
2. Most simultaneous equations systems are over-identified.
 
Indirect Least Squares (ILS)

20. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•In fact, we can use this technique for just-identified and over-identified systems.
 
•Two stage least squares (2SLS or TSLS) is done in two stages:
 
Stage 1:
•Obtain and estimate the reduced form equations using OLS. Save the fitted
values for the dependent variables.
 
Stage 2:
•Estimate the structural equations, but replace any RHS endogenous variables
with their stage 1 fitted values.
 
Estimation of Systems
Using Two-Stage Least Squares

21. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Example: Say equations (14)-(16) are required.
 
Stage 1:
•Estimate the reduced form equations (17)-(19) individually by OLS and obtain the
fitted values, .
 
Stage 2:
•Replace the RHS endogenous variables with their stage 1 estimated values:
 
(24)-(26)
 
•Now and will not be correlated with u
1
, will not be correlated with u
2
, and
will not be correlated with u
3
.
Estimation of Systems
Using Two-Stage Least Squares (cont’d)
,,YYY
123
Y Y Y X Xu
Y Y Xu
Y Yu
1 0 12 33 41 52 1
2 0 13 21 2
3 0 12 3
    
  
 



 


Y
2
Y
3
Y
3
Y
2

22. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•It is still of concern in the context of simultaneous systems whether the
CLRM assumptions are supported by the data.
•If the disturbances in the structural equations are autocorrelated, the
2SLS estimator is not even consistent.
 
•The standard error estimates also need to be modified compared with
their OLS counterparts, but once this has been done, we can use the
usual t- and F-tests to test hypotheses about the structural form
coefficients.
Estimation of Systems
Using Two-Stage Least Squares (cont’d)

23. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Recall that the reason we cannot use OLS directly on the structural equations is that the
endogenous variables are correlated with the errors.
 
•One solution to this would be not to use Y
2
or Y
3
, but rather to use some other variables
instead.
 
•We want these other variables to be (highly) correlated with Y
2
and Y
3
, but not correlated
with the errors - they are called INSTRUMENTS.
 
•Say we found suitable instruments for Y
2
and Y
3
, z
2
and z
3
respectively. We do not use the
instruments directly, but run regressions of the form
 
(27) & (28)
 
Instrumental Variables
Y z
Y z
2 1 22 1
3 3 43 2
 
 



24. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Obtain the fitted values from (27) & (28), and , and replace Y
2
and
Y
3
with these in the structural equation.
•We do not use the instruments directly in the structural equation.
 
•It is typical to use more than one instrument per endogenous variable.
 
•If the instruments are the variables in the reduced form equations, then
IV is equivalent to 2SLS.
Instrumental Variables (cont’d)
Y
2
Y
3

25. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
What Happens if We Use IV / 2SLS Unnecessarily?
•The coefficient estimates will still be consistent, but will be inefficient
compared to those that just used OLS directly.
 
The Problem With IV
•What are the instruments?
Solution: 2SLS is easier.
 
Other Estimation Techniques
 

1. 3SLS - allows for non-zero covariances between the error terms.
2. LIML - estimating reduced form equations by maximum likelihood
3. FIML - estimating all the equations simultaneously using maximum
likelihood
Instrumental Variables (cont’d)

26. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•George and Longstaff (1993)
•Introduction
- Is trading activity related to the size of the bid / ask spread?
- How do spreads vary across options?
 
•How Might the Option Price / Trading Volume and the Bid / Ask Spread be
Related?
 
Consider 3 possibilities:
1. Market makers equalise spreads across options.
2. The spread might be a constant proportion of the option value.
3. Market makers might equalise marginal costs across options irrespective
of trading volume.
An Example of the Use of 2SLS: Modelling
the Bid-Ask Spread and Volume for Options

27. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•The S&P 100 Index has been traded on the CBOE since 1983 on a
continuous open-outcry auction basis.
•Transactions take place at the highest bid or the lowest ask.
 
•Market making is highly competitive.
Market Making Costs

28. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•For every contract (100 options) traded, a CBOE fee of 9c and an
Options Clearing Corporation (OCC) fee of 10c is levied on the firm
that clears the trade.
 
•Trading is not continuous.
 
•Average time between trades in 1989 was approximately 5 minutes.
What Are the Costs Associated with Market Making?

29. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•The CBOE limits the tick size:
 
$1/8 for options worth $3 or more
$1/16 for options worth less than $3
 
•The spread is likely to depend on trading volume
... but also trading volume is likely to depend on the spread.
 
•So there will be a simultaneous relationship.
 
 
The Influence of Tick-Size Rules on Spreads

30. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•All trading days during 1989 are used for observations.
 
•The average bid & ask prices are calculated for each option during the
time 2:00pm – 2:15pm Central Standard time.
 
•The following are then dropped from the sample for that day:
 

1. Any options that do not have bid / ask quotes reported during the ¼ hour.
2. Any options with fewer than 10 trades during the day.
 
•The option price is defined as the average of the bid & the ask.
 
•We get a total of 2456 observations. This is a pooled regression.
The Data

31. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•For the calls:
 (1)
 
(2)
 
•And symmetrically for the puts:
(3)
 
(4)
 
where PR
i
& CR
i
are the squared deltas of the options
 
The Models
CBA CDUM C CL T CRe
i i i i i ii
       
0 1 2 3 4 5
CL CBA T T Mv
i i i i i i
     
0 1 2 3
2
4
2
PBA PDUM P PL T PRu
i i i i i i i
       
0 1 2 3 4 5
PL PBA TT Mw
i i i i i i    
0 1 2 3
2
4
2

32. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•CDUM
i
and PDUM
i
are dummy variables
= 0 if C
i
or P
i
< $3
= 1 if C
i
or P
i
 $3
 
•T
2
allows for a nonlinear relationship between time to maturity and the spread.
 
•M
2
is used since ATM options have a higher trading volume.
 
•Aside: are the equations identified?
•Equations (1) & (2) and then separately (3) & (4) are estimated using 2SLS.
The Models (cont’d)

33. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Results 1

C a l l B i d -A s k S p r e a d a n d T r a d i n g V o l u m e R e g r e s s i o n

iiiiiii eC RTCLCCDUMCB A 
543210  ( 6 . 5 5 )
iiiiii vMTTCB ACL 
2
4
2
3210  ( 6 . 5 6 )


0

1

2

3

4

5 A d j . R
2

0 . 0 8 3 6 2
( 1 6 . 8 0 )
0 . 0 6 1 1 4
( 8 . 6 3 )
0 . 0 1 6 7 9
( 1 5 . 4 9 )
0 . 0 0 9 0 2
( 1 4 . 0 1 )
-0 . 0 0 2 2 8
(-1 2 . 3 1 )
-0 . 1 5 3 7 8
(-1 2 . 5 2 )
0 . 6 8 8

0

1

2

3

4 A d j . R
2

-3 . 8 5 4 2
(-1 0 . 5 0 )
4 6 . 5 9 2
( 3 0 . 4 9 )
-0 . 1 2 4 1 2
(-6 . 0 1 )
0 . 0 0 4 0 6
( 1 4 . 4 3 )
0 . 0 0 8 6 6
( 4 . 7 6 )
0 . 6 1 8
N o t e : t-r a t i o s i n p a r e n t h e s e s . S o u r c e : G e o r g e a n d L o n g s t a f f ( 1 9 9 3 ) . R e p r i n t e d w i t h t h e p e r m i s s i o n o f
t h e S c h o o l o f B u s i n e s s A d m i n i s t r a t i o n , U n i v e r s i t y o f W a s h i n g t o n .

34. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Results 2

P u t B i d -A s k S p r e a d a n d T r a d i n g V o l u m e R e g r e s s i o n
iiiiiii
uP RTP LPP DUMP B A 
543210
 ( 6 . 5 7 )
iiiiii wMTTP B AP L 
2
4
2
3210  ( 6 . 5 8 )

0

1

2

3

4

5 A d j . R
2

0 . 0 5 7 0 7
( 1 5 . 1 9 )
0 . 0 3 2 5 8
( 5 . 3 5 )
0 . 0 1 7 2 6
( 1 5 . 9 0 )
0 . 0 0 8 3 9
( 1 2 . 5 6 )
-0 . 0 0 1 2 0
(-7 . 1 3 )
-0 . 0 8 6 6 2
(-7 . 1 5 )
0 . 6 7 5

0

1

2

3

4 A d j . R
2

-2 . 8 9 3 2
(-8 . 4 2 )
4 6 . 4 6 0
( 3 4 . 0 6 )
-0 . 1 5 1 5 1
(-7 . 7 4 )
0 . 0 0 3 3 9
( 1 2 . 9 0 )
0 . 0 1 3 4 7
( 1 0 . 8 6 )
0 . 5 1 7
N o t e : t-r a t i o s i n p a r e n t h e s e s . S o u r c e : G e o r g e a n d L o n g s t a f f ( 1 9 9 3 ) . R e p r i n t e d w i t h t h e p e r m i s s i o n o f
t h e S c h o o l o f B u s i n e s s A d m i n i s t r a t i o n , U n i v e r s i t y o f W a s h i n g t o n .

35. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Adjusted R
2
 60%
 

1
and 
1
measure the tick size constraint on the spread
 

2
and 
2
measure the effect of the option price on the spread
 

3
and 
3
measure the effect of trading activity on the spread
 

4
and 
4
measure the effect of time to maturity on the spread
 

5
and 
5
measure the effect of risk on the spread
 

1
and 
1
measure the effect of the spread size on trading activity etc.
Comments:

36. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•The paper argues that calls and puts might be viewed as substitutes
since they are all written on the same underlying.
•So call trading activity might depend on the put spread and put trading
activity might depend on the call spread.
 
•The results for the other variables are little changed.
Calls and Puts as Substitutes

37. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Bid - Ask spread variations between options can be explained by reference to the
level of trading activity, deltas, time to maturity etc. There is a 2 way relationship
between volume and the spread.
 
•The authors argue that in the second part of the paper, they did indeed find
evidence of substitutability between calls & puts.
 
Comments
 - No diagnostics.
 - Why do the CL and PL equations not contain the CR and PR variables?
 - The authors could have tested for endogeneity of CBA and CL.
- Why are the squared terms in maturity and moneyness only in the
liquidity regressions?
 - Wrong sign on the squared deltas.
Conclusions

38. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•A natural generalisation of autoregressive models popularised by Sims
•A VAR is in a sense a systems regression model i.e. there is more than one
dependent variable.
 
•Simplest case is a bivariate VAR
where u
it
is an iid disturbance term with E(u
it)=0, i=1,2; E(u
1t u
2t)=0.
 
•The analysis could be extended to a VAR(g) model, or so that there are g
variables and g equations.
Vector Autoregressive Models
y y y y y u
y y y y y u
t t ktk t ktk t
t t ktk t ktk t
1 10 1111 11 1121 12 1
2 20 2121 22 2111 21 2
    
    
   
   
   
   
... ...
... ...

39. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•One important feature of VARs is the compactness with which we can write
the notation. For example, consider the case from above where k=1.
 
•We can write this as
 
or
 
 
or even more compactly as
 
y
t
= 
0
+ 
1
y
t-1
+ u
t
g1 g1 gg g1 g1
Vector Autoregressive Models:
Notation and Concepts
y y y u
y y y u
t t t t
t t t t
1 10 1111 1121 1
2 20 2121 2111 2
  
  
 
 
 
 
y
y
y
y
u
u
t
t
t
t
t
t
1
2
10
20
11 11
21 21
11
21
1
2





































40. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•This model can be extended to the case where there are k lags of each
variable in each equation:
y
t
= 
0
+ 
1
y
t-1
+ 
2
y
t-2
+...+ 
k
y
t-k
+ u
t
g1 g1 gg g1gg g1 gg g1 g1
•We can also extend this to the case where the model includes first
difference terms and cointegrating relationships (a VECM).
Vector Autoregressive Models:
Notation and Concepts (cont’d)

41. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
•Advantages of VAR Modelling
- Do not need to specify which variables are endogenous or exogenous - all are endogenous
- Allows the value of a variable to depend on more than just its own lags or combinations
of white noise terms, so more general than ARMA modelling
- Provided that there are no contemporaneous terms on the right hand side of the equations, can
simply use OLS separately on each equation
- Forecasts are often better than “traditional structural” models.
•Problems with VAR’s
- VAR’s are a-theoretical (as are ARMA models)
- How do you decide the appropriate lag length?
- So many parameters! If we have g equations for g variables and we have k lags of each of the
variables in each equation, we have to estimate (g+kg
2
) parameters. e.g. g=3, k=3, parameters =
30
- Do we need to ensure all components of the VAR are stationary?
- How do we interpret the coefficients?
Vector Autoregressive Models Compared with
Structural Equations Models

42. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Choosing the Optimal Lag Length for a VAR

2 possible approaches: cross-equation restrictions and information criteria
Cross-Equation Restrictions

In the spirit of (unrestricted) VAR modelling, each equation should have
the same lag length

Suppose that a bivariate VAR(8) estimated using quarterly data has 8 lags
of the two variables in each equation, and we want to examine a restriction
that the coefficients on lags 5 through 8 are jointly zero. This can be done
using a likelihood ratio test

Denote the variance-covariance matrix of residuals (given by /T), as .
The likelihood ratio test for this joint hypothesis is given by
ˆ
uuˆˆ
 
urTLR 
ˆ
log
ˆ
log

43. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Choosing the Optimal Lag Length for a VAR
(cont’d)
where is the variance-covariance matrix of the residuals for the restricted
model (with 4 lags), is the variance-covariance matrix of residuals for the
unrestricted VAR (with 8 lags), and T is the sample size.
•The test statistic is asymptotically distributed as a 
2
with degrees of freedom
equal to the total number of restrictions. In the VAR case above, we are
restricting 4 lags of two variables in each of the two equations = a total of 4 *
2 * 2 = 16 restrictions.
•In the general case where we have a VAR with p equations, and we want to
impose the restriction that the last q lags have zero coefficients, there would
be p
2
q restrictions altogether
•Disadvantages: Conducting the LR test is cumbersome and requires a
normality assumption for the disturbances.
r

ˆ
u
ˆ

44. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Information Criteria for VAR Lag Length Selection
• Multivariate versions of the information criteria are required. These can
be defined as:
where all notation is as above and k is the total number of regressors in
all equations, which will be equal to g
2
k + g for g equations, each with k
lags of the g variables, plus a constant term in each equation. The values
of the information criteria are constructed for 0, 1, … lags (up to some
pre-specified maximum ).k
ln(ln(T))
2
ˆln
ln(T)ˆln
/2ˆln
T
k
MHQIC
T
k
MSBIC
TkMAIC







45. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Does the VAR Include Contemporaneous Terms?
•So far, we have assumed the VAR is of the form
 
•But what if the equations had a contemporaneous feedback term?
 
•We can write this as
•This VAR is in primitive form.
y y y u
y y y u
t t t t
t t t t
1 10 1111 1121 1
2 20 2121 2111 2
  
  
 
 
 
 
y y y yu
y y y yu
t t t t t
t t t t t
1 10 1111 1121 122 1
2 20 2121 2111 221 2
   
   
 
 
  
  
y
y
y
y
y
y
u
u
t
t
t
t
t
t
t
t
1
2
10
20
11 11
21 21
11
21
12
22
2
1
1
20
0


















































46. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Primitive versus Standard Form VARs
•We can take the contemporaneous terms over to the LHS and write
or
B y
t
= 
0
+ 
1
y
t-1
+ u
t
 
•We can then pre-multiply both sides by B
-1
to give
 y
t
= B
-1

0
+ B
-1

1
y
t-1
+ B
-1
u
t
or
y
t
= A
0
+ A
1
y
t-1
+ e
t
•This is known as a standard form VAR, which we can estimate using OLS.
1
1
22
12 1
2
10
20
11 11
21 21
11
21
1
2






































 



y
y
y
y
u
u
t
t
t
t
t
t

47. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Block Significance and Causality Tests
• It is likely that, when a VAR includes many lags of variables, it will be
difficult to see which sets of variables have significant effects on each
dependent variable and which do not. For illustration, consider the following
bivariate VAR(3):
• This VAR could be written out to express the individual equations as

• We might be interested in testing the following hypotheses, and their
implied restrictions on the parameter matrices:



















































































t
t
t
t
t
t
t
t
t
t
u
u
y
y
y
y
y
y
y
y
2
1
32
31
2221
1211
22
21
2221
1211
12
11
2221
1211
20
10
2
1








tttttttt
tttttttt
uyyyyyyy
uyyyyyyy
2322231212222212112221121202
1321231112212211112121111101







48. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Block Significance and Causality Tests (cont’d)
•Each of these four joint hypotheses can be tested within the F-test framework,
since each set of restrictions contains only parameters drawn from one equation.
•These tests could also be referred to as Granger causality tests.
•Granger causality tests seek to answer questions such as “Do changes in y
1

cause changes in y
2?” If y
1 causes y
2, lags of y
1 should be significant in the
equation for y
2
. If this is the case, we say that y
1
“Granger-causes” y
2
.
•If y
2
causes y
1
, lags of y
2
should be significant in the equation for y
1
.
•If both sets of lags are significant, there is “bi-directional causality”
Hypothesis Implied Restriction
1. Lags of y1t do not explain current y2t

21 = 0 and

21 = 0 and

21 = 0
2. Lags of y1t do not explain current y1t

11 = 0 and

11 = 0 and

11 = 0
3. Lags of y2t do not explain current y1t

12 = 0 and

12 = 0 and

12 = 0
4. Lags of y2t do not explain current y2t

22 = 0 and

22 = 0 and

22 = 0

49. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Impulse Responses
•VAR models are often difficult to interpret: one solution is to construct
the impulse responses and variance decompositions.
•Impulse responses trace out the responsiveness of the dependent variables
in the VAR to shocks to the error term.
 A unit shock is applied to each
variable and its effects are noted.
•Consider for example a simple bivariate VAR(1):
•A change in u
1t
will immediately change y
1
. It will change change y
2
and
also y
1 during the next period.
•We can examine how long and to what degree a shock to a given
equation has on all of the variables in the system.
y y y u
y y y u
t t t t
t t t t
1 10 1111 1121 1
2 20 2121 2111 2
  
  
 
 
 
 

50. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Variance Decompositions
•Variance decompositions offer a slightly different method of examining
VAR dynamics. They give the proportion of the movements in the
dependent variables that are due to their “own” shocks, versus shocks to the
other variables.

•This is done by determining how much of the s-step ahead forecast error
variance for each variable is explained innovations to each explanatory
variable (s = 1,2,…).
•The variance decomposition gives information about the relative
importance of each shock to the variables in the VAR.

51. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Impulse Responses and Variance Decompositions:
The Ordering of the Variables
•But for calculating impulse responses and variance decompositions, the ordering
of the variables is important.
•The main reason for this is that above, we assumed that the VAR error terms
were statistically independent of one another.
•This is generally not true, however. The error terms will typically be correlated
to some degree.
•Therefore, the notion of examining the effect of the innovations separately has
little meaning, since they have a common component.
•What is done is to “orthogonalise” the innovations.
•In the bivariate VAR, this problem would be approached by attributing all of the
effect of the common component to the first of the two variables in the VAR.
•In the general case where there are more variables, the situation is more complex
but the interpretation is the same.

52. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
An Example of the use of VAR Models:
The Interaction between Property Returns and the
Macroeconomy.
•Brooks and Tsolacos (1999) employ a VAR methodology for investigating the
interaction between the UK property market and various macroeconomic variables.
•Monthly data are used for the period December 1985 to January 1998.
•It is assumed that stock returns are related to macroeconomic and business conditions.
•The variables included in the VAR are
–FTSE Property Total Return Index (with general stock market effects removed)
–The rate of unemployment
–Nominal interest rates
–The spread between long and short term interest rates
–Unanticipated inflation
–The dividend yield.
The property index and unemployment are I(1) and hence are differenced.

53. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Marginal Significance Levels associated with Joint
F-tests that all 14 Lags have not Explanatory Power
for that particular Equation in the VAR
•Multivariate AIC selected 14 lags of each variable in the VAR
Lags of Variable
Dependent variableSIR DIVY SPREADUNEM UNINFLPROPRES
SIR 0.00000.00910.02420.03270.21260.0000
DIVY 0.50250.00000.62120.42170.56540.4033
SPREAD 0.27790.13280.00000.43720.65630.0007
UNEM 0.34100.30260.11510.00000.07580.2765
UNINFL 0.30570.51460.34200.47930.00040.3885
PROPRES 0.55370.16140.55370.89220.72220.0000

54. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Variance Decompositions for the
Property Sector Index Residuals
•Ordering for Variance Decompositions and Impulse Responses:
–Order I: PROPRES, DIVY, UNINFL, UNEM, SPREAD, SIR
–Order II: SIR, SPREAD, UNEM, UNINFL, DIVY, PROPRES.
Explained by innovations in
SIR DIVY SPREAD UNEM UNINFL PROPRES
Months ahead I II I II I II I II I II I II
1 0.00.80.038.20.0 9.10.00.7 0.0 0.2100.051.0
2 0.20.80.235.10.212.30.41.4 1.6 2.9 97.547.5
3 3.82.50.429.40.217.81.01.5 2.3 3.0 92.345.8
4 3.72.15.322.31.418.51.61.1 4.8 4.4 83.351.5
12 2.83.115.58.715.319.53.35.117.013.546.150.0
24 8.26.36.8 3.938.036.25.514.718.116.923.422.0

55. ‘Introductory Econometrics for Finance’ © Chris Brooks 2008
Impulse Responses and Standard Error Bands for
Innovations in Dividend Yield and
the Treasury Bill Yield
Innovations in Dividend Yields
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
123456789101112131415161718192021222324
Steps Ahead
Innovations in the T-Bill Yield
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
123456789101112131415161718192021222324
Steps Ahead